4 Structural health monitoring of CFRP via electrical resistance
4.3 Results
4.3.4 Analytical model
To gain a deeper understanding of the influence of IFF and delaminations on the electrical resistance change of CFRP, an analytical model is proposed in the following.
The analytical model is developed to be able to predict the electrical resistance change for defined loading and corresponding damage states of a CFRP specimen with printed conductive paths (configuration shown in Figure 4.5). With the model, an identification of the contributions to the electrical resistance change of both IFF and delaminations shall be possible.
Furthermore, with such a model, the electrical resistance changes can be calculated for different laminate layups and electrical conductivity values and therefore also for other material systems.
Considering the anisotropic electrical resistances of the system, an equivalent circuit diagram is developed for the CFRP specimen (see Figure 4.21). The total resistance is measured from one to the other end of the printed path. It comprises the electrical resistance of the printed path (RP), the contact resistance between printed path and CFRP (RC), the resistance in thickness direction for each layer (Rt), and the 0°‐layers and 90°‐layers resistances in path direction for each layer (R0 and R90). The model is based on the assumption that damages reduce the effective area that can be used for electron transport, leading to a local increase of the electrical resistance.
For this model, the following assumptions and simplifications are made:
The material exhibits ohmic behavior. Hence, Ohm’s law applies.
Electrical current density is homogeneously distributed in all cross sections.
Electrical resistance is the same for all layers of the same fiber direction and same damage state (i.e. R0 = R0,1 = R0,2 = R0,3 = R0,4 and R90,lower = R90,1 = R90,2 = R90,3 and R90,upper = R90,4 = R90,5 = R90,6).
Electrical resistance in thickness direction between two adjacent plies is the same, independent of the fiber directions of the two neighboring layers, but it decreases for damaged layers (i.e.
Rt,lower = Rt,1 = Rt,2 = Rt,3 and Rt = Rt,4 = Rt,5 = Rt,6 = Rt,7 = Rt,8 = Rt,9).
The contact resistance between printed path and the neighboring layer is half of the resistance as the one between two neighboring layers, because the resin rich inter‐laminar layer between path and surface is half the size compared to the inter‐laminar layer between two adjacent layers.
In a first step, we consider the influence of IFF only. Therefore, we assume that after the first IFF, the effective areas of the printed path and of the lower 90°‐layers are zero because the IFF interrupt these Figure 4.21: (a) Layup of the CFRP laminate with printed conductive path; (b)
Equivalent circuit diagram of the system.
Structural health monitoring of CFRP via electrical resistance measurement
With the experimentally determined dimensions of a printed path, the effective area of the printed path can be expressed as
700 ∙ 10 ∙ 8 ∙ 10
0 0
0 (4.5)
Where i is the number of IFF.
The effective areas in thickness direction are
, , ,
15 ∙ 10 ∙ 50 ∙ 10
15 ∙ 10 ∙ 50 ∙ 10 ∙ ∙ 0
0 (4.6)
, , , , , ,
15 ∙ 10 ∙ 50 ∙ 10 (4.7)
Moreover, in the 90°‐ and 0°‐layers, the effective areas can be expressed as
, , ,
0.1875 ∙ 10 ∙ 15 ∙ 10 0
0 0
(4.8)
, , , , , , ,
0.1875 ∙ 10 ∙ 15 ∙ 10 (4.9)
The lengths for calculating the resistances are taken from the specimen elements depicted in Figure 4.21 (b) are calculated using Ohm’s law:
∙ , , 90, 0 (4.12)
With the resistances and the equivalent circuit diagram, the total resistance Rtotal) for the initial state and for any number of IFF (i) can be expressed as:
1
Structural health monitoring of CFRP via electrical resistance measurement
The initial electrical resistance from the experiments is 2.47 ± 0.24 Ω and the analytical model calculates 2.37 Ω. Hence, the model gives a slightly lower initial resistance than observed experimentally, but the value is in the scatter of the experimental values (some measured values lie above and some lie below the calculated values).
To compare the analytical results with results from the electrical measurements, electrical resistance changes of the first 10 IFF are analyzed for 14 different experimental measurements. For the first calculation, the reduction of the effective area (p) is kept constant. This means that we assume the change of effective area to be independent of the number of IFF (except for the first IFF – here, the effective area is not reduced) as well as independent of the position of the failure. Calculations are conducted for p = 5 % to p = 10 % and plotted with the experimental data (see Figure 4.22).
These values were chosen as they fit well into the range of the corresponding resistance increases per IFF observed in the experiments.
Figure 4.22: Resistance change for first ten inter‐fiber failures with assumption that every inter‐fiber failure has the same influence on the resistance change,
1 2 3 4 5 6 7 8 9 10
For low numbers of IFF, i < 5, the analytical solutions for p > 7 fit best to the experimental values. However, for higher numbers of IFF, i > 5, these analytical solutions overestimate the resistance change significantly and the analytical solutions for p = 6 and p = 7 lie closer to the measured values. In general, for increasing number of IFF, the analytical solutions overestimate the experimentally determined values for the resistance change. This discrepancy is caused by the fact that statistically the reduction of the effective area decreases for increasing number of IFF, because of two reasons: First, if an IFF occurs between two existing IFF, the resistance increase is significantly smaller. Second, the more cracks are present, the closer the cracks lie together and the effect of the change of the effective area decreases. Therefore, it is necessary to adjust the model as described in the next paragraph.
Due to the statistically decreasing influence of a crack for increasing numbers of IFF, the parameter p is adjusted depending on the number of IFF (i) as follows:
(4.14)
0.94 ∙ for i > 0 (4.15)
The factor 0.94 is determined from the experiments by evaluation of the decreasing resistance changes for increasing number of IFF.
Structural health monitoring of CFRP via electrical resistance measurement
In addition to the influence of IFF, the change of the effective area due to delaminations needs to be considered. The delamination area is taken from the ultrasonic C‐scans (see Figure 4.18) and plotted in a diagram with a linear fit describing the delamination area. The axis describing the number of IFF is also approximated from the experimental results (see Figure 4.23).
Due to the different x‐axes of the number of IFF and the displacement, the linear fit function needs to be defined. Then, the delamination area can be expressed as follows:
0
101.86 ∙ 10 14.82 ∙ 10 ∙ 6
7 (4.16)
Figure 4.23: Delamination area from ultrasonic C‐scans versus displacement and
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 20 40 60 80 100 120 140 160 180 200
linear fit function experiment
Delamination area in mm²
Displacement in mm 1 2 3 4 5 6 7 8 9 10
Number of inter-fiber failures
Therefore, the effective area of the lower 90°‐layers considering both IFF and delaminations is:
, , ,
15 ∙ 10 ∙ 50 ∙ 10
15 ∙ 10 ∙ 50 ∙ 10 ∙ ∙ 0
0 (4.17)
With the addition of the above described equations, the results of the analytical model are plotted in Figure 4.24.
1 2 3 4 5 6 7 8 9 10
0 20 40 60 80 100
p = 5 p = 6 p = 7 p = 8 p = 9 p = 10 experiment
Resistance changeR/R0 in %
Structural health monitoring of CFRP via electrical resistance measurement
It can be seen that the analytical results lie close to the experimental results for all regarded damage states for p = 9 and p = 10. The best results are obtained for p = 9.5 (see Figure 4.25). In addition, Figure 4.25 shows the contributions to the electrical resistance change caused by IFF and delaminations that were calculated separately. The influence of the IFF dominates but the curve for the IFF shows a slightly regressive behavior.
Delaminations contribute to the electrical resistance change from the 7th IFF and the curve shows a progressive behavior. Therefore, it can be concluded that the influence of IFF dominates for the less damaged state. However, for increasing amount of damages, the influence of the delaminations increases.
The reason why the contribution of delaminations to the resistance increase sets in after several IFF are already present can be explained by the fact that delaminations occur after the first IFF because of high inter‐laminar shear stresses at the crack tips of the present IFF. Eventually, the delaminations start growing leading to an increasing contribution of the delaminations.
Figure 4.25: Resistance change calculated by analytical model for parameter p = 9.5 and experimental results as well as contribution of inter‐fiber failures
1 2 3 4 5 6 7 8 9 10 p = 9.5 contribution of IFF + delaminations contribution of IFF
contribution of delaminations experimental
The analytical model shows good agreement with experiments for the initial resistance of the undamaged specimen. Furthermore, the analytical results show good agreement with experiments for electrical resistance change caused by IFF and delaminations. The model can be used to predict the electrical resistance change for a defined loading and damage state and it can be estimated how the IFF and delaminations each contribute to the total electrical resistance change. With relatively small adjustments, this analytical model could be adapted to other cross‐ply laminate layups with different geometries and electrical conductivities.
Structural health monitoring of CFRP via electrical resistance measurement